Quadratic Functions

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Quadratic Functions · Grade 9 · Worksheet 1

1. Mason is designing a custom skateboard ramp for a competition. The ramp's cross-section follows the path y = 2x² - 16x + 35, where y is the height in inches and x is the horizontal distance in feet from the starting point. What is the minimum height of the ramp in inches? Answer: 3 Solution: Identify the coefficients from y = 2x² - 16x + 35, where a = 2, b = -16, and c = 35. Find the x-coordinate of the vertex using x = -b/(2a) = -(-16)/(2*2) = 16/4 = 4.
   Full step-by-step solution

   Step 1: Identify the coefficients from y = 2x² - 16x + 35, where a = 2, b = -16, and c = 35. Step 2: Find the x-coordinate of the vertex using x = -b/(2a) = -(-16)/(2*2) = 16/4 = 4. Step 3: Substitute x = 4 back into the equation to find the minimum height: y = 2(4)² - 16(4) + 35. Step 4: Calculate: y = 2(16) - 64 + 35 = 32 - 64 + 35 = 3. Step 5: The minimum height of the ramp is 3 inches.

2. Emma is designing a custom frame for her art project. The frame's border follows the quadratic function y = 2x² - 12x + 19, where y represents the border width in centimeters at position x along the frame. What is the minimum width of the border in centimeters? Answer: 1 Solution: Start with the quadratic function: y = 2x² - 12x + 19 To find the minimum value, convert to vertex form by completing the square Factor out the coefficient of x²: y = 2(x² - 6x) + 19 Complete the square inside the parentheses: x² - 6x = (x² - 6x + 9) - 9 = (x - 3)² - 9 Substitute back: y = 2[(x…
   Full step-by-step solution

   Step 1: Start with the quadratic function: y = 2x² - 12x + 19 Step 2: To find the minimum value, convert to vertex form by completing the square Step 3: Factor out the coefficient of x²: y = 2(x² - 6x) + 19 Step 4: Complete the square inside the parentheses: x² - 6x = (x² - 6x + 9) - 9 = (x - 3)² - 9 Step 5: Substitute back: y = 2[(x - 3)² - 9] + 19 Step 6: Distribute the 2: y = 2(x - 3)² - 18 + 19 Step 7: Simplify: y = 2(x - 3)² + 1 Step 8: The vertex form shows the vertex is at (3, 1) Step 9: Since the coefficient 2 is positive, the parabola opens upward, so the vertex gives the minimum value Step 10: The minimum width is the y-coordinate of the vertex: 1 cm The answer is 1.

3. Matiu is designing a custom skateboard ramp. The shape of the ramp follows the quadratic function y = 2(x - 4)² + 6, where y represents the height in inches and x represents the horizontal distance in feet from the starting edge. To calculate the ramp's dimensions at ground level, Matiu needs to convert this equation to standard form. What is the coefficient of the x term in the standard form of this quadratic equation? Answer: -16 Solution: Start with the vertex form: y = 2(x - 4)² + 6 Expand the squared binomial: (x - 4)² = x² - 8x + 16 Multiply by the coefficient 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Add the constant 6: y = 2x² - 16x + 32 + 6 = 2x² - 16x + 38 Identify the coefficient of the x term: -16 The answer is -16.
   Full step-by-step solution

   Step 1: Start with the vertex form: y = 2(x - 4)² + 6 Step 2: Expand the squared binomial: (x - 4)² = x² - 8x + 16 Step 3: Multiply by the coefficient 2: 2(x² - 8x + 16) = 2x² - 16x + 32 Step 4: Add the constant 6: y = 2x² - 16x + 32 + 6 = 2x² - 16x + 38 Step 5: Identify the coefficient of the x term: -16 The answer is -16.

4. Kaia is designing a custom skateboard ramp that follows a parabolic path. The height of the ramp (in centimeters) is modeled by the quadratic function y = 4(x - 11)² + 96. Kaia needs to convert this equation to standard form y = ax² + bx + c to program her design software. What is the value of the coefficient b in the standard form equation? Answer: -88 Solution: Start with the vertex form: y = 4(x - 11)² + 96 Expand the squared binomial: (x - 11)² = (x - 11)(x - 11) = x² - 22x + 121 Multiply by the coefficient 4: 4(x² - 22x + 121) = 4x² - 88x + 484 Add the constant 96: y = 4x² - 88x + 484 + 96 Combine like terms: y = 4x² - 88x + 580 The coefficient b is…
   Full step-by-step solution

   Step 1: Start with the vertex form: y = 4(x - 11)² + 96 Step 2: Expand the squared binomial: (x - 11)² = (x - 11)(x - 11) = x² - 22x + 121 Step 3: Multiply by the coefficient 4: 4(x² - 22x + 121) = 4x² - 88x + 484 Step 4: Add the constant 96: y = 4x² - 88x + 484 + 96 Step 5: Combine like terms: y = 4x² - 88x + 580 Step 6: The coefficient b is the number multiplied by x, which is -88 Therefore, the value of b is -88.

5. Matiu is analyzing the flight path of a drone using the quadratic function y = -2x² + 20x - 32, where y represents the drone's height in meters and x represents the horizontal distance in meters. What is the maximum height the drone reaches during its flight? Answer: 18 Solution: Identify the coefficients: a = -2, b = 20, c = -32 Find the x-coordinate of the vertex using x = -b/(2a) x = -20/(2×(-2)) = -20/(-4) = 5 Substitute x = 5 into the original equation to find the maximum height y = -2(5)² + 20(5) - 32 y = -2(25) + 100 - 32 y = -50 + 100 - 32 y = 50 - 32 y = 18 The…
   Full step-by-step solution

   Step 1: Identify the coefficients: a = -2, b = 20, c = -32 Step 2: Find the x-coordinate of the vertex using x = -b/(2a) x = -20/(2×(-2)) = -20/(-4) = 5 Step 3: Substitute x = 5 into the original equation to find the maximum height y = -2(5)² + 20(5) - 32 y = -2(25) + 100 - 32 y = -50 + 100 - 32 y = 50 - 32 y = 18 Step 4: The maximum height is 18 meters.

6. Sophia is designing a triangular garden with vertices at coordinates (0,0), (8,0), and (4,12). She wants to place a decorative fountain at the centroid of this triangle. What is the y-coordinate of the centroid? Answer: 4 Solution: The centroid of a triangle is found by averaging the x-coordinates and y-coordinates of its three vertices separately. This geometric principle applies to any triangle regardless of its shape or orientation.
   Full step-by-step solution

   The centroid of a triangle is found by averaging the x-coordinates and y-coordinates of its three vertices separately. For example, if you had a triangle with vertices (1,2), (3,4), and (5,6), you would calculate the centroid by finding the average of the x-values (1+3+5)/3 and the average of the y-values (2+4+6)/3. This geometric principle applies to any triangle regardless of its shape or orientation.

7. Convert y = 3(x - 8)² + 9 to standard form Answer: y = 3x² - 48x + 201 Solution: Start with y = 3(x - 8)² + 9 Expand (x - 8)² = x² - 16x + 64 Multiply by 3: 3(x² - 16x + 64) = 3x² - 48x + 192 Add the constant term: 3x² - 48x + 192 + 9 Combine like terms: 3x² - 48x + 201 Final answer: y = 3x² - 48x + 201
   Full step-by-step solution

   Step 1: Start with y = 3(x - 8)² + 9 Step 2: Expand (x - 8)² = x² - 16x + 64 Step 3: Multiply by 3: 3(x² - 16x + 64) = 3x² - 48x + 192 Step 4: Add the constant term: 3x² - 48x + 192 + 9 Step 5: Combine like terms: 3x² - 48x + 201 Final answer: y = 3x² - 48x + 201